Communication complexity of quasirandom rumor spreading

Abstract

We consider rumor spreading on random graphs and hypercubes in the quasirandom phone call model. In this model, every node has a list of neighbors whose order is specified by an adversary. In step i every node opens a channel to its ith neighbor (modulo degree) on that list, beginning from a randomly chosen starting position. Then, the channels can be used for bi-directional communication in that step. The goal is to spread a message efficiently to all nodes of the graph. We show three results. For random graphs (with sufficiently many edges) we present an address-oblivious algorithm with runtime O(logn) that uses at most O(n log log n) message transmissions. For hypercubes of dimension log n we present an address-oblivious algorithm with runtime O(log n) that uses at most O(n (log log n)2) message transmissions. For hypercubes we also show a lower bound of Ω(n log n/log log n) on the total number of message transmissions required by any O(log n) time address-oblivious algorithm in the standard random phone call model. Together with a result of [8], our results imply that for random graphs and hypercubes the communication complexity of the quasirandom phone call model is significantly smaller than that of the standard phone call model. This seems to be surprising given the small amount of randomness used in our model. © 2010 Springer-Verlag.